Top 10 Logical Paradoxes

Paradoxes are more than just unexplained phenomena. Some people call it a “paradox” that we have some evidence suggesting certain stars are the same age and other evidence suggesting they’re different ages, but that’s not any kind of seeming logical impossibility… it’s like noticing that one fraternal twin has crow’s feet, probably because he works harder than the other one.

And not all paradoxes are created equal. Some are way too specific, some show shallow thinking, some are restatements of basic ideas found in other paradoxes, and some appear to have been created with the aid of psychotropic drugs (“Dude. Why do we park in the driveway… AND DRIVE IN THE PARKWAY?” “Dude. What’s a ‘parkway?'”). Here, then, are the ones we’ve found most beguiling over the years.

10. Schrodinger’s Cat. The cool cat Schroeder has been put in a closed box by the jealous Lucy. Or something like that. You can’t see or hear the cat. Because of the observer effect, the cat is literally both alive and dead at the same time. Seen in many, many books, TV shows, movies, video games.Dresden Codak‘s treatment is a personal favorite.

9. The Tree in the Forest. If it falls, does it make a sound? How, exactly, do you define “sound?” This, and many paraphrases, figure into lots and lots of arguments, often as a synonym for the observer effect. The idea sometimes translates into fantasy as gods, fairies or entire worlds in danger of fading from existence if their “audiences” no longer hear them. For some fantasy/sci-fi writers, that may be a little too autobiographical to be healthy.

8. Achilles and the Tortoise. The tortoise had a head start in the race. Achilles is catching up. He keeps halving the distance between himself and the tortoise… but will never quite reach it, because adding 1/2, then 1/4, then 1/8 and so on ad infinitum will get you closer and closer to the number 1, but never quite there. The answer to this one will give you a firmer understanding of math in the real world, especially factorials. Seen in: informal calculus, the terrible romcom I.Q.

6. The Paradox of the Stone. Can God create one so heavy He can’t lift it? If his power is infinite, what happens when infinity is turned against itself? It’s generally assumed that whether He can or can’t, He doesn’t… except in a few works of fiction and a few mythologies like the Greek, wherein the god/God who created the world finds himself running scared from his own creations like a divine Frankenstein. Seen in: Preacher.

5. The Nihilist’s Challenge. Not its official name, but, fittingly, it doesn’t really have one. Nihilists do not believe anything exists– and defy you to prove otherwise, which is insanely difficult since their thesis challenges any axiom you might use to begin your reasoning. It’s such a challenge that led Rene Descartes to “Cogito ergo sum…” which nihilists didn’t take long to start arguing against. A softer form of nihilism declares the world to be real, but meaningless, or to have no meaning save that which we impose (which is basically Sartre’s Existentialism). Seen in: Rorschach’s origin story and implied in the pitch for Seinfeld,“the show about nothing.”

4. Escher Stairs. We could have picked one of a half-dozen Escher artworks to represent his distinctive, logically challenging artwork, a form of visual paradox. But people always seem most taken with those stairs, perhaps because they’re so immersive. Seen in the movies Harry Potter and the Sorceror’s Stone and Inception, among others.

2. The Liar Paradox. “This sentence is a lie.” This statement is neither true nor false, merely dickish. The classic A.I.-destroyer, elegant in its simplicity. Seen in: one unforgettable Star Trek sequence.

1. Time Travel. What happens if you go back in time and try to kill your own grandfather? If all time fits together like a jigsaw, as in The Time Traveler’s Wife, then everything is predetermined, including your journey, and you’ll be prevented by some narratively frustrating coincidence. This seems pretty tough to believe, but arguably even harder to swallow is the idea that doing this would spark a wildfire through the fabric of the universe, destroying everything, as claimed in the Back To The Future series. The best-sounding theory (branching parallel universes) still conjures up images of another Earth being created every time a time traveler draws a breath. It may be that our heads are not currently equipped to deal with the true principles of time travel, just as the rules of quantum mechanics mock our intuition. But we’re learning those…

Discussion (10) ¬

Paradoxes take three forms; and two of them are not actually paradoxes. At least, according to Will Quine. They’re either: truth masquerading as absurdity (‘veridical’ paradoxes), falsehood masquerading as mystery (‘falsidical’) and… something else. Something called an antinomy. A statement for which no truth or falsehood can be applied; when reason is applied, it defies both.

The only veridical paradox in your list is schrodinger’s cat. The rest are falsidical, except for Russell’s paradox, and under certain interpretations the Liar’s Paradox. Those two are the only ‘real’ paradoxes on your list, from a mathematical and educated philosopher’s perspective: the antinomy.

The problem with antimonies is this: how do you know it’s *actually* a true paradox? For example, Xeno’s paradox would have been considered an antimony for a long time, until mathematicians finally trounced it. Likewise, the Liar’s Paradox can usually be explained away as a failing in the English language that allows ludicrous constructions, unless you’re extremely rigorous in setting up the terms of the reasoning applied to it.

Russell’s Paradox is interesting, because it cannot be explained away. As an extension to it, Goedel’s incompleteness theorem actually proves that not only is it impossible to explain away Russell’s, but that EVERY formal system of logic will produce inexplicable paradoxes that invalidate the originating system.

Mathematicians who dabble in proof theory HATE that one. Their latest gambit to make an end-run around Goedel is something called the ‘Axiom of Choice’, which basically states that even though it seems to be mathematically impossible to create a set of rules that allow choosing one item exclusively out of each set within an infinitity set of ‘random’ sets,we must be able to do it anyway, because… we do. That ability to choose one from many is something that we as thinking beings seem to be able to do; therefore, it must be possible.

From there, and following an arcane series of logical steps, proof theorists have come up with ways to create situations where Goedel’s theorem does not come into play.

But personally, I find great fault in the Axiom of choice. It seems like a bit of a cop out; like the old problem of seeing the sun rise and assuming that it must move around the earth, instead of the earth revolving.

In other words, who’s to say that we’re actually choosing ANYTHING from an infinite set? Infinity is large; and for it to be truly infinite, it must contain something which limits it. If it did not, it wouldn’t be truly infinite; only conditionally infinite. And if it did, then, it would not be infinite. Yet for this situation to arise, infinity must first exist in order to be invalidated. It’s Russell’s paradox all over again, given a different form.

For the most part, I just want to bow to your obviously vastly superior knowledge of paradox theory. But a couple questions do spring to mind.

Is the Paradox of the Stone not useful even to atheist (or agnostic) mathematicians grappling with the issue of infinity? Considered in that light, would it be more veridical than falsidical?

And what’s the failing in the English language you speak of? It seems like self-reflexivity is an important concept the language would be poorer without, yet its inclusion necessitates the Liar’s Paradox or some variation thereof.

Sure, it’s useful. It’s a falsidical statement that proves the problem of omnimpotence, using a thought experiment.

That doesn’t make it a paradox, unless somehow not being able to make a stone too heavy to life means you /must/ be able to make a stone too heavy to lift.

The failing in the English language is this: English is imprecise. That’s why we have “math”; symbolic logic can do things that conversational vocabulary can not.

However, self-reflexivity is a good point to bring up, because it’s key to Godel’s incompleteness. He proved that it is impossible to build a complete system of logical rules without self-reference; and that the feedback loops are then guaranteed to create a situation where the self referential statement is ‘I’m wrong’. The liar’s paradox is a good way to illustrate this in easy language, but the flaw is that it’s too credible to argue that “This statement” — i.e., the self-reference — refers to an instance of the statement, and not its logical construction as a whole.

Another way to think of it is like a man turning a crank. You have to sit there and feed the question, “is this true?” into the statement “this is a lie”, to get your first contradictory result.

Then, if you ask, “Is this a lie”, you get the /opposite/ result.

But you only get one answer per question. There’s no feedback loop without you participating.

That’s the problem with a paradox using conversational language; it only produces a ‘self’ denial if you’re there to hold a mirror to it. Otherwise, it’s just words.

However, Russell’s paradox doesn’t have that problem, since it relates to a practical difficulty of anyone who is trying to create a proof using set theory. Can’t get around it by not reading it, in other words.

Oh, your question was if the stone paradox would be veridical. Not really; the construction is based on a false premise – “If omnipotence exists, then is it possible to create a stone too heavy to lift, and still be omnipotent? If you can’t, are you then omnipotent?” And is used to prove the negative, i.e., omnipotence without qualifiers is not possible.

You could construct the words in such a way to avoid statements that are false, by assuming the answer is ‘no omnipotence’ and working backwards, but because the premise it’s countering is still proven untrue, it’s considered falsidical.

Don’t forget paradoxes that arise when one assumes that every statement is either 100% true or false, and which require a continuum of truth in order to resolve. E.g. the Sorites (Heap) paradox:

A pile of, let’s say, a million grains of sand surely constitutes a ‘heap’ of sand. (If you disagree, then say a billion, or whatever you would agree would be a heap.)

Now, we can surely agree that a single grain of sand does not constitute a ‘heap’ by normal definitions of the word. (If you disagree, then the paradox disappears, but only because you’re using the word differently than most people do.)

If you add another grain of sand to the single grain of sand, then it does not become a heap (again using a reasonable definition of ‘heap’) and if you take away one of the million grains of sand in a heap, it doesn’t stop being a heap, unless you arbitrarily define a ‘heap’ as being exactly one million grains of sand or more. So we can conclude from this that adding or taking away one grain of sand is not sufficient to change the ‘heapness’ status of a grouping of sand.

So if you add one more grain of sand to the two grains, it’s three grains… still not a heap… add one more, four grains, still not a heap… see where I’m going with this? Logically, all you’re doing is adding one more grain at a time, so at no time does the group of sand gain the status of a heap, because as we noted before, adding one grain of sand does not change the status, and in order for a group of sand to become a heap, it would have to do so as a result of an action we took during the process, and the only action is adding one grain at a time, so adding one million grains of sand one at a time is not sufficient to create a heap! On the other hand, there is no meaningful difference in a million grains of sand if they are put together one at a time or a lot at once, so from this we would have to conclude that a heap of sand cannot, in fact, exist.

Of course, in a more well-rounded logical system we can say that two grains of sand is slightly more ‘heap-like’ than one grain of sand, and that three grains is a little more than that, and so forth, until somewhere between one and one million one could say, depending on how you define things, that a threshold level of heap-ness is reached and now you have a heap.

person 1: “i like everything you like”
person 2: “i hate everything you like and like what you hate”

so the first person likes so 2 likes nothing so person 1 likes nothing so 2 likes everything ext

also this ones nice

by doing nothing your doing somthing but your doing nothing but doing nothing is doing somthing by doing nothing because nothing is somthing. so you can do nothing but that is something so you cant do nothing because nothing is something so you can do something by doing something

I’d say the first is a Russell paradox, and the second basically boils down to the statement “nothing is something,” a philosophical paradox which doesn’t seem to line up too closely with any of my ten. I’d check the links and parahacker’s comments for further insight.

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